Dynamic control of the permanent magnet assisted reluctance synchronous machine with constant current angle

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(1)Dynamic Control of the Permanent Magnet Assisted Reluctance Synchronous Machine with Constant Current Angle. Hugo Werner de Kock. Thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Electronic Engineering with Computer Science at the University of Stellenbosch. Promoter: Prof. Maarten J. Kamper, University of Stellenbosch. March 2006.

(2) Declaration: I, the undersigned, hereby declare that the work contained in this thesis is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree.. H.W. de Kock. Signature. March 2006 Stellenbosch, South Africa. ii.

(3) Abstract This thesis is about the dynamic control of a permanent magnet assisted reluctance synchronous machine (PMA RSM). The PMA RSM in this thesis is a 110 kW traction machine and is ideal for the use in electrical rail vehicles. An application of the dynamic control of the PMA RSM in electrical rail vehicles is to reduce wheel slip.. The mathematical model of the PMA RSM is derived and explained in physical terms. Two methods of current control for the PMA RSM are investigated, namely constant field current control (CFCC) and constant current angle control (CCAC). It is shown that CCAC is more appropriate for the PMA RSM.. A current controller for the PMA RSM that guarantees stability is derived and given as an analytic formula. This current controller can be used for any method of current control, i.e. CFCC or CCAC. An accurate simulation model for the PMA RSM is obtained using results from finite element analysis (FEA). The accurate model is used in a simulation to verify CCAC. A normal proportional integral speed controller for the PMA RSM is designed and the design is also verified by simulation.. Practical implementation of the current and speed controllers is considered along with a general description of the entire drive system. The operation of the resolver (for position measurement) is given in detail. Important safety measures and the design of the electronic circuitry to give protection are shown. Practical results concerning current and speed control are then shown.. To improve the dynamic performance of the PMA RSM, a load torque observer with compensation current feedback is investigated. Two observer structures are considered, namely the reduced state observer and the full state observer. The derivation of the full state observer and the detail designs of the observer elements are given. The accurate simulation model of the PMA RSM is used to verify the operation of the observer structures and to evaluate the dynamic performance.. Both observer. structures are implemented practically and practical results are shown.. One method of position sensorless control, namely the high frequency voltage injection method, is discussed in terms of the PMA RSM. This work is additional to the thesis but it is shown, because it raises some interesting questions regarding the dynamic control of the PMA RSM.. iii.

(4) Opsomming Hierdie tesis gaan oor die dinamiese beheer van ’n permanente-magneet-ondersteunde reluktansie sinchroon-masjien (PMO RSM). Die PMO RSM in hierdie tesis is ’n 110 kW trekkrag masjien en is ideaal vir die gebruik in elektriese spoor voertuie. ’n Toepassing van die dinamiese beheer van die PMO RSM in elektriese spoor voertuie is om wiel glip te verminder.. ’n Wiskundige model van die PMO RSM word afgelei en verduidelik in terme van fisiese begrippe. Twee metodes van stroombeheer vir die PMO RSM, naamlik konstante vloedstroom beheer (KVB) en konstante stroomhoek beheer (KSB), word ondersoek. Daar word aanbeveel dat KSB vir die PMO RSM gebruik word.. ’n Stroombeheerder wat stabiliteit verseker word vir die PMO RSM afgelei en gegee as ’n analitiese formule. Hierdie stroombeheerder kan gebruik word vir enige tipe stroombeheer, d.i. vir KVB of KSB. Eindige element analise word gebruik om ’n akkurate simulasiemodel van die PMO RSM te verkry. Hierdie simulasiemodel word dan gebruik om die werking van KSB te bevestig. ’n Normale proportioneel-integraal spoedbeheerder word ontwerp en die werking van die spoedbeheerder word bevestig deur simulasie.. Praktiese implementering van die stroom- en spoedbeheerders word beskryf en ’n geheelbeeld van die totale beheerstelsel word getoon. Die werking van die ‘resolver’ (vir posisie meting) word in detail verduidelik. Belangrike beveiligingstappe en die ontwerp van elektronika wat die beveiliging daarstel, word gegee. Praktiese stroom- en spoedmetings word ook getoon.. In ’n poging om die dinamiese gedrag van die PMO RSM te verbeter, word ’n draaimomentafskatter met kompensasiestroom-terugvoer ondersoek. Twee tipes afskatters, naamlik die vereenvoudigdetoestand-afskatter en die vol-toestand-afskatter word bestudeer. Die afleiding van die vol- toetstandafskatter en die detail ontwerp van al die afskatter-elemente word gegee. Die akkurate simulasiemodel van die PMO RSM word gebruik om die werking van beide afskatters te toets. Beide afskatters word prakties geïmplementeer en die praktiese resultate word getoon.. Een metode van posisie-sensorlose-beheer, naamlik die hoë-frekwensie spanningsinjeksie metode word bespreek in terme van die PMO RSM. Hierdie werk is bykomend tot die tesis, maar word getoon omdat dit interesante vrae aangaande die dinamiese beheer van die PMO RSM uitlok.. iv.

(5) Acknowledgements I would like to express my sincere appreciation to: •. My promoter, Prof. Maarten Kamper, for his assistance, advice and encouragement.. •. Spoornet and the University of Stellenbosch for financial assistance.. •. The International Office at the University of Stellenbosch for assistance regarding my exchange visit to the University of Wuppertal, Germany.. •. Mr. André Cabral Ferreira at the University of Wuppertal for his assistance regarding sensorless control.. •. Mr. Aniel le Roux for designing the digital signal processor and writing firmware for it and also for his assistance during initial stages of the control program development.. •. My friend, Mr. Richard Brady, for his support, encouragement and advice.. •. Mr. Francois Rossouw and Mr. André Swart for their technical support regarding the practical test setup.. •. Mr. Yongle Ai, Mr. Edward Rakgati, Mr. Arnold Rix and Dr. Rong-Jie Wang for their’ support and assistance.. •. My parents and my sister for their endless love and support.. v.

(6) “I know this world is ruled by infinite intelligence. Everything that surrounds us – everything that exists – proves that there are infinite laws behind it. There can be no denying this fact. It is mathematical in its precision.”. Thomas A. Edison 1847 – 1931. vi.

(7) Glossary of symbols and abbreviations These symbols and abbreviations are in no particular order. In some equations uppercase letters are used – this refers to the steady state.. Symbol. Meaning. Unit. φ. magnetic flux. Weber [Wb]. current angle. degrees or radians. δ. flux linkage angle. degrees or radians. i. current. Ampere [A]. e. induced voltage. Volt [V]. v. voltage. Volt [V]. T. torque. [kg.m2]. r or R. resistance. Ohm [Ω]. ω. rotational speed. Radians per second [rad/sec]. L. self-inductance. Henry [H]. M. mutual inductance. Henry [H]. J. inertia. Newton second [Ns]. B. friction coefficient. Newton metre second per radian. λ. magnetic flux linkage. Weber turns [Wb turns]. p. pole pairs. scalar. P. power. Watt [W]. d dt. time derivative. scalar. ∂ ∂t. partial time derivative. scalar. θ. rotor position. degrees or radians. power factor angle. degrees or radians. γ. inner power factor angle. degrees or radians. j. square root of -1. scalar. KT. torque coefficient. Newton metre per ampere [Nm/A]. or. or. vii.

(8) Abbreviation. Meaning. DC. direct current. RSM. reluctance synchronous machine. AC. alternation current. PMA RSM. permanent magnet assisted RSM. FEA. finite element analysis. PM. permanent magnet. d-axis. direct axis. q-axis. quadrature axis. Fig.. figure. ABC. stationary reference frame. QD0. synchronously rotating reference frame. CFCC. constant field current control. CCAC. constant current angle control. rms. root mean squared. ZOH. zero order hold. S-plane. continuous plane. Z-plane. discrete plane. W-plane. bilinear transform plane. LHP. left hand plane. RHP. right hand plane. Z{…}. the Z-transform. DSP. digital signal processor. ADC. analogue-to-digital converter. DAC. digital-to-analogue converter. LPF. low pass filter. LUT. lookup table. PI. proportional integral. P. proportional. DSP. digital signal processor. FPGA. field programmable gate array. ISR. interrupt service routine. IGBT. insulated gate bipolar transistor. viii.

(9) Table of Contents Chapter 1. Introduction ........................................................................................................... 1. 1.1 Background ....................................................................................................................... 1 1.2 Problem Statement ............................................................................................................ 7 1.3. Approach to the problem................................................................................................... 8 1.4. Thesis layout...................................................................................................................... 9. Chapter 2. Mathematical model for the PMA RSM ............................................................. 10. 2.1 The mathematical model for the RSM ............................................................................ 10 2.2 The mathematical model for the PMA RSM................................................................... 17 2.3. Power factor and inner power factor ............................................................................... 19 2.4. Parameter values.............................................................................................................. 20 2.5. Method of current control for RSM and PMA RSM....................................................... 24. Chapter 3. Design methodology for the current and speed controllers................................. 30. 3.1. Different methods to design a digital controller.............................................................. 30 3.2. Decoupling procedure for the electrical model ............................................................... 31 3.3. Open loop current response............................................................................................. 33 3.4. Closed loop current response .......................................................................................... 35 3.5. The discrete nature of the controller................................................................................ 36 3.6. Design in the W-plane using a Bode plot and gain margin specification ....................... 39 3.7. Revised current controllers.............................................................................................. 44 3.8. Implementation of the current controllers in simulation ................................................. 45 3.9. The inverter model and the modified electrical machine model ..................................... 49 3.10. Design of the speed controller......................................................................................... 53 3.11. Implementation of the speed controller in simulation..................................................... 56 3.12. Simulation results for the speed controller...................................................................... 60 3.13. Noise in the measured speed signal................................................................................. 63 3.14. An alternative speed controller........................................................................................ 63 3.15. Summary for design methodology .................................................................................. 64. ix.

(10) Chapter 4. Practical considerations....................................................................................... 65. 4.1. Representation of the machine as seen from the load ..................................................... 66 4.2. The rectifier ..................................................................................................................... 67 4.3. The dumping circuit and the warning light circuit .......................................................... 68 4.4. The load system............................................................................................................... 73 4.5. The inverter ..................................................................................................................... 73 4.6. DC bus voltage and three-phase current measurement ................................................... 75 4.7. Rotor angular position measurement using a resolver .................................................... 78 4.8. Zero position.................................................................................................................... 85 4.9. Overview of the DSP....................................................................................................... 88 4.10. The control program........................................................................................................ 89 4.11. Practical measurements ................................................................................................... 92. Chapter 5. Load torque observer........................................................................................... 95. 5.1. Why is the load torque observer needed?........................................................................ 95 5.2. Load torque observer for RSM with CFCC .................................................................... 98 5.3. Reduced state observer for RSM (or PMA RSM) with CCAC..................................... 103 5.4. Full state observer for RSM (or PMA RSM) with CCAC ............................................ 114 5.5. Summary for load torque observer with compensation current .................................... 129. Chapter 6. Position sensorless control ................................................................................ 130. 6.1. High frequency voltage injection .................................................................................. 131 6.2. Analysis of the current response and the rotor angle tracking scheme ......................... 134. Chapter 7. Summary, conclusions and recommendations .................................................. 139. 7.1. Conclusions ................................................................................................................... 139 7.2. Recommendations ......................................................................................................... 140. x.

(11) Appendix R. References ..................................................................................................... 141. R.1. Articles on CDROM...................................................................................................... 141 R.2. Books............................................................................................................................. 142 R.3. P-CAD 2000 circuit diagram on CDROM .................................................................... 143 R.4. Datasheets / Brochures on CDROM ............................................................................. 143 R.6. Matlab 7 analysis on CDROM ...................................................................................... 144 R.7. Programs on CDROM ................................................................................................... 144 R.8. Matlab 7 simulation files on CDROM .......................................................................... 144 R.9. Videos on CDROM ....................................................................................................... 145 R.10. Websites ........................................................................................................................ 145. Appendix A. Torque equation............................................................................................. 146. Appendix B. Zero order hold and the Z-transform............................................................. 150. Appendix C. The rotating magnetic field ........................................................................... 156. Appendix D. Park transform in the time domain ................................................................ 158. Appendix E. Lookup tables ................................................................................................ 162. Appendix F. Signal to noise ratio....................................................................................... 174. Appendix G. Full state observer ......................................................................................... 175. Appendix H. Additional simulation results......................................................................... 179. xi.

(12) Chapter 1. Introduction. This chapter provides some background information to orientate the reader with regard to the field of electrical machines and drives. The problem, which is addressed in this thesis, will then be presented. An overview of the problem solving approach and the thesis layout follows.. 1.1. Background. Many types of electrical machines that can act as motors or generators exist today; these include amongst others the DC machine, induction machine and wound rotor synchronous machine. These machines have been studied and perfected in their design over many years. One machine that did not receive much attention is the reluctance synchronous machine (RSM)1, because it was known to a have very poor performance.. This is true when the machine is controlled in an open-loop manner.. However, by controlling the machine in a closed loop with feedback, its performance is comparable with other popular machines. Indeed, the RSM has recently received renewed attention [A8], mainly due to the modern field orientated control strategies [A9][A10][A21].. In the market of general-purpose drives, the induction motor still represents a standard as an off-theshelf motor [A11], but there is now a shift towards synchronous motor based drives due to guaranteed efficiency. To be a competitive product in terms of general-purpose applications, a drive must be both low-cost and well suited to sensorless control2. Research shows that the RSM is a good candidate [A11][A12].. A very basic explanation of the electrical machines mentioned above will now be given for the sole purpose of orienting the reader. It is interesting to note that the separately excited DC motor and the RSM can be controlled in the same way. This will be pointed out in the text to follow.. 1 2. Many authors use the term “synchronous reluctance machine” This is control without a physical position (or speed) sensor such as a resolver or encoder. 1.

(13) The DC machine There are two types of DC machines namely permanent magnet DC machines, which give a constant field flux φ f , and field winding DC machines, which give a variable field flux according to a field current φ f = k f i f [B1, p.99]. The DC machine has the armature windings in the rotor in contrast to the AC machine which has the armature windings in the stator. The DC machine gives mechanical rectification of the AC voltage that is induced in its rotor. The field winding of the DC machine may be connected in series with the armature winding, or the DC machine may be excited separately.. The machine equations for the separately excited DC machine are as follows:. Tem = k tφ f ia. (1.1.1). ea = k t φ f ω m. (1.1.2). vt = ea + Ra ia + L'a Tem = J. dia dt. dω m + Bω m + Twl (t ) dt. (1.1.3) (1.1.4). In the equations above the symbols have the following meaning:. Tem is the electromagnetic torque produced by the interaction between the armature current ia and the field flux φ f . The back-EMF ea induced in the rotor, is proportional to the field flux and the rotor speed ω m . To establish the armature current, a terminal voltage vt is applied. The steady state value of the armature current is determined according to equation (1.1.3) by the applied terminal voltage, back-EMF and the stator winding resistance Ra . The rate of change of the armature current is limited by the self inductance L'a . The rotor speed is a function of the produced electromagnetic torque, the load torque Twl , the inertia of the rotor J and the friction coefficient B .. k t is a constant of. proportionality.. 2.

(14) In a separately excited DC motor, the field flux is produced by a separately controlled field current. Note that in the equivalent circuit for the separately excited DC motor, shown in Fig. 1.1, the circuits for the field and armature are separated.. Rf. + Vf -. Lf. Ra. + Vt -. flux. If La. + Ea. Ia. Fig. 1.1. Equivalent circuit of the separately excited DC motor.. In the steady state, with equation (1.1.2) substituted into equation (1.1.3), the separately excited DC motor can be described by the following equations:. Tem = k t φ f I a. (1.1.5). Vt = k eφ f ω m + Ra I a. (1.1.6). Vf = Rf I f. (1.1.7). ωm =.  R 1  Vt − a Tem   k eφ f  ktφ f . (1.1.8). From equation (1.1.8) it is clear that the speed can be increased linearly by keeping both field and torque constant3 and increasing the terminal voltage linearly. In this range the armature current and torque are kept constant at their rated values, therefore it is known as the constant torque region.. If the terminal voltage has reached its rated value, the speed can be further increased by reducing the field4 in the machine and keeping the terminal voltage constant. In this region constant torque cannot be maintained because that would require the armature current in equation (1.1.5) to increase above its rated value. The torque will therefore have to decrease as the speed increases beyond the rated speed. This region is known as the constant power region, since both current and voltage are kept constant. Fig. 1.2 shows (in per unit quantities) the two regions mentioned.. 3 4. This is called “constant field control” This is called “field weakening”. 3.

(15) Per Unit. Tem , ia , φ f , i f. v t , ia. 1.0. ea Tem , φ f , i f. vt ea 1.0. ωm Per Unit. Fig. 1.2. Constant torque and constant power regions for the DC motor.. The figure above is with reference to the separately excited DC motor, but this graph can be produced for any kind of electrical motor and generally follows the same patterns. It is usually the aim of electrical machine designers to make the constant power region large and to minimize the loss in torque in the constant power region.. The induction machine By introducing three-phase power, it is possible to create a rotating magnetic field (this is explained thoroughly in Appendix C). The induction machine [B1, p.263] uses three-phase current to create a rotating magnetic field in its stator. The rotating magnetic field induces voltage in the rotor and if current is allowed to flow (for instance by connecting the rotor to external resistors), then there are current carrying conductors in a magnetic field and thus torque is produced. The produced torque causes rotation to occur. It is important to note that the rotor does not rotate at the same speed as the rotating magnetic field (synchronous speed): the rotor lags behind the rotating field in motor operation and the rotating field lags behind the rotor in generator operation.. Fig. 1.3. Induction machine with wound rotor and external resistors. The wound rotor synchronous machine Injecting a current into the rotor, such that the magnetic poles created by this action can lock with the rotating magnetic field poles due to the three-phase stator currents, causes the rotor to turn at synchronous speed. This is the basic operation of a wound rotor synchronous machine.. 4.

(16) Fig. 1.4. Wound rotor synchronous machine. The reluctance synchronous machine If the rotor is magnetically symmetrical and no currents can flow in the rotor, then it will not follow the rotating magnetic field created by the three-phase stator currents (in fact it will not turn at all). If the rotor is magnetically asymmetrical5 however, it will follow the rotating magnetic field exactly due to the reluctance force. The rotor will therefore turn at synchronous speed. This is the basic operation of a reluctance synchronous machine (RSM).. Fig. 1.5. Two pole RSM. From the explanations above, it is clear that the rotor design of the RSM is much different from both the induction machine and wound rotor synchronous machine, while the stator design is very similar. Comparing the RSM and DC machine, the RSM has the advantage that it has no brushes or rotor windings, thus requiring much less maintenance. Control of the RSM is simpler because there are no rotor parameters to be identified [B2, p.1]. The cooling of the RSM is also easier.. Comparative studies carried out on the performance of the RSM and induction machine in the lowand medium-power range show that the RSM has higher torque density and efficiency [A2][A3][A4]. It was found that the RSM has more advantages than the induction machine, because the absence of rotor currents leads to a simple vector control scheme and insignificant rotor losses. The rotor manufacturing cost for the RSM is relatively low compared to that of the induction machine.. 5. Also referred to as “magnetic anisotropy” or “magnetic saliency”. 5.

(17) Even though feedback control increases the performance of the RSM dramatically, there are still some inherent disadvantages: it has a lower power factor and its performance deteriorates fast in the fluxweakening region compared to the induction machine [A1]. To address this problem, permanent magnets (PMs) can be placed inside the flux barriers of the reluctance rotor. These drives are defined as PM-assisted RSM drives [A5][A6][A7].. This thesis is concerned with the control of a permanent magnet assisted reluctance synchronous machine (PMA RSM). The rated power of the machine is 110 kW. The rotor was first designed to be a normal RSM [B17] and was then upgraded to be a PMA RSM [B5]. In both cases finite element analysis (FEA) with an optimization procedure [B2, p.8] was used. A graphical representation of the optimally designed PMA RSM is shown in Fig. 1.6. From Fig. 1.6 it is clear that the PM sheets are curved. A picture of the actual PM sheets is shown in Fig. 1.7.. Fig. 1.6. Optimally designed cross section segment of the PMA RSM.. Fig. 1.7. PM sheets that are placed inside the rotor flux barriers of the RSM.. 6.

(18) A cross-section of the rotor with the PM sheets in the flux barriers is shown in Fig. 1.8. From this figure it is clear that there is a direction from the centre outward that has no flux barriers in its path: this is called the direct axis (D-axis). Flux can travel freely in this direction and is only limited by the saturation of the steel. The other direction has flux barriers in its path, which makes it difficult for the flux to travel: this is called the quadrature axis (Q-axis). In fact, there are three D-axes and three Qaxes, because this is a six-pole rotor. This rotor is magnetically asymmetrical, because flux travels freely in the one direction, but not in the other.. Q-axis. D-axis. Fig. 1.8. Complete six-pole rotor of PMA RSM.. 1.2. Problem Statement. The problem is to get the best dynamic performance from the PMA RSM, but at the same time to ensure energy efficient operation. Dynamic performance is the ability of the machine to follow the speed reference as closely as possible under varying load conditions. The load, for example, might fall away suddenly in which case the motor will accelerate quickly. It is then the aim of the controller to bring the motor speed back to the reference speed as fast as possible. The torque equation of the RSM (see Appendix A) is given by:. 3 Tem = ( ) p( Ld − Lq ) ⋅ id ⋅ iq 2. (1.2.1). By keeping the D-axis current constant in (1.2.1), the D-axis self inductance is also constant. The Qaxis inductance is almost constant, since there is almost no saturation of flux in this direction. In this method, the torque is therefore only a function of the Q-axis current. This means that the machine is controlled in a constant field manner, just like the separately excited DC motor.. Constant field control can be viewed as a standard method of controlling the RSM. This method is however not energy efficient since the field current (D-axis current) is always present even under no. 7.

(19) load conditions.. A comparative study [B3] shows that the constant current angle method of. controlling the RSM is more energy efficient, giving maximum torque per ampere, but results in slightly slower dynamic performance. This can be explained briefly: the rate of change of current is limited by self inductance, i.e. a greater self inductance results in a lower rate of change of current. The D-axis self inductance is greater than the Q-axis self inductance and therefore the D-axis current response is slower than the Q-axis current response. With constant field control, it is only the Q-axis current that needs to change and therefore this method gives a fast current (and torque) response. With constant current angle control however, both Q- and D-axis currents are being changed, so this means the total response time is the slowest response time, which is the D-axis current response time.. The question is therefore “what method of current control should be used?” The author investigates the possibilities and proposes additional methods to increase the response time of the machine. It was also stated that in order to be competitive in general-purpose applications, the machine needs to be controlled without a position sensor. One method of position sensorless control for the PMA RSM is explained in concept, although this is not the main focus of the thesis.. 1.3.. Approach to the problem. A theoretical model for the PMA RSM is derived and the viability of constant field current control and constant current angle control is then explored. An accurate model for the PMA RSM is obtained using results from finite element analysis (FEA). It is then possible to design stable current controllers and a speed controller. Simulation is used to verify these controllers. The controllers are implemented practically and practical results are compared to the simulation results.. The load torque observer with compensation current feedback is then introduced as a way to improve the dynamic performance of the machine drive. Again, theoretical development is verified with simulation and compared to practical results. Finally, one method of position sensorless control for the PMA RSM is presented.. 8.

(20) 1.4.. Thesis layout. The layout of the remainder of this thesis is as follows:. Chapter 2:. A model for the PMA RSM is derived theoretically and FEA is used to obtain the actual variables, namely flux linkages, self inductances and mutual inductances. The advantages of the PMs are discussed. Finally, the viability of constant field control and constant current angle control is investigated.. Chapter 3:. The design methodology for stable current controllers and the speed controller is presented. The operation of the controllers is verified by simulation.. Chapter 4:. Practical considerations are presented together with practically measured results of the current and speed control.. Chapter 5:. Theoretical development of the load torque observer with compensation current feedback, as a method of increasing dynamic response, is presented. Simulation results are compared with practically measured results.. Chapter 6:. Position sensorless control using high frequency voltage injection is presented with respect to the PMA RSM. A simulation block diagram is shown, although the detail designs of all the elements in the block diagram are not shown, nor any simulation results.. Practical results for the method applied to a small RSM in Wuppertal,. Germany is shown to prove that the method works.. Chapter 7:. In this chapter a summary with conclusions is given and recommendations are made for further research.. 9.

(21) Chapter 2. Mathematical model for the PMA RSM. In this chapter a mathematical model for the RSM will be developed, which is followed by a mathematical model for the PMA RSM. A critical evaluation of the current control method for both machines is then performed.. The mathematical machine model is necessary to understand how the machine works and to be able to construct a simulation model.. Simulation results will eventually be compared with practically. measured results. If the simulation model is accurate, the results should be very similar.. The. simulation of the controlled PMA RSM is discussed in Chapter 3.. To understand the physical difference between the RSM and the PMA RSM, consider Fig. 2.1. In the figure, a cross-section of the RSM (left) and a cross-section of the RSM with permanent magnet sheets incorporated into the reluctance rotor (right), are shown. The permanent magnets have certain beneficial effects which are described later in this chapter.. Flux barrier. PM sheet. Fig. 2.1. Cross section of the RSM (left) and PMA RSM (right).. 2.1. The mathematical model for the RSM. Since the rotor of the RSM does not contain any windings, the electrical model is based only on the stator. The induced stator voltage is given by Faraday’s Law and the copper loss component also has to be taken into account in the voltage equation as follows:. vabc =. dλabc + rs iabc dt. (2.1.1). 10.

(22) The equation above is in the stationary ABC reference frame which is fixed to the stator. The stator windings are represented as three stationary coils displaced by 120º spatially. The model of the machine can be simplified by using Park’s transform (see Appendix A). The voltage equations in the QD0 reference frame, which is fixed to the rotor, are given by:. dλ d − ω e λq dt. v d = rs id + v q = rs iq +. dλ q dt. + ω e λd. (2.1.2). (2.1.3). The zero-component is indeed zero when balanced three-phase voltages are applied, so it is omitted here. It is important to gain physical insight into equations (2.1.2) and (2.1.3). The rest of this subsection explains these equations as well as the torque equation in detail.. The Park transform changes the voltage equations from a stationary ABC reference frame to a synchronously rotating QD0 reference frame. It is a one-to-one mapping that is dependent on the rotor position. Note that the QD0 equations are still as seen from the stator. The stator is now represented by two coils displaced by 90º spatially. The two stator coils rotate synchronously with the rotor. This representation is shown in Fig. 2.1.1. magnetic Q-axis. D'. λq Q'. Q. λd. magnetic D-axis. D. Fig. 2.1.1. Park transform representation of stator coils.. Fig. 2.1.1 is a two-dimensional representation of a three-dimensional object. In this representation there are two coils to consider. Each coil lies flat on its own plane. The two planes are perpendicular to each other and also perpendicular to the page. Therefore, id flows in a coil that lies on a vertical plane (the D-plane), perpendicular to the page and iq flows in a coil that lies on a horizontal plane (the Q-plane) that is also perpendicular to the page. A current in a coil produces magnetic flux linkage that moves through the area spanned by the coil in a direction given by the right hand rule. For example, if. id flows in the direction indicated in Fig. 2.1.1 (in at the cross and out at the dot), then flux linkage moves perpendicularly through the D-plane in the direction of the magnetic D-axis.. 11.

(23) The currents and flux linkages can be displayed together on a phasor diagram. In phasor diagrams, peak values are always used to indicate magnitude and not root mean squared (rms) values. The QD0 currents and flux linkages are DC quantities however, and therefore the notion of peak values or rms values is not applicable (see Appendix D). In this thesis, the unit for the currents in the QD0 reference frame are therefore simply [A] and not [A rms] or even [A peak]. In the phasor diagram id and λ d are in the same direction, although it is known that the current actually moves in a plane and the flux linkage moves through that plane. It is convenient to speak of the “D-axis current” and the “D-axis flux linkage”. The phasor diagram Fig. 2.1.2, shows the D-axis current, D-axis flux linkage, Q-axis current, Q-axis flux linkage, stator current, stator flux linkage, current angle and flux angle. Q-axis. is. iq λq. φ. λs δ. λd. D-axis. id. Fig. 2.1.2. Vector diagram for currents and flux linkages in the QD0 reference frame.. In Fig. 2.1.2, the stator current, is , is the vector sum of the D-axis current and Q-axis current components. It should be noted that the magnitude of the stator current gives the peak amplitude of the currents in the ABC reference frame (see Appendix D). Therefore, if the rated current of the machine is irated = 200 A rms = 283 A peak , then is rated = 283 A .. Using Fig. 2.1.1 and Fig. 2.1.2, the D-axis voltage equation is given by v d = rs ⋅ id +. (. ). d (λd + jλq ) in dt. the time domain, and by v d = rs ⋅ id + s ⋅ λ d + jλ q in the Laplace domain. This equation should be evaluated at the appropriate frequency, which in this case is the electrical rotational speed of the machine. ωe .. Therefore. v d = rs ⋅ id + jω e ⋅ (λ d + jλ q ) ,. which. can. be. written. as. v d = rs ⋅ id + jω e λ d − λ q ω e .. 12.

(24) Transforming back to the Laplace domain: v d = rs ⋅ id + s ⋅ λ d − λ q ω re and finally to the time domain: v d = rs ⋅ id +. d λd − λq ω re . This is not a physical explanation of equation (2.1.2), but it dt. gives a mathematical explanation for the negative speed voltage term − λ q ω re . Using Fig. 2.1.1 and Fig. 2.1.2, the Q-axis voltage equation is given by v q = rs ⋅ iq +. d (λq − jλd ) . Following the same dt. reasoning as for the D-axis voltage equation, the Q-axis voltage equation (2.1.3) can be derived.. 3 2. Appendix A shows that the torque equation for the RSM is given by Tem = ( ) p (λ d ⋅ iq − λ q ⋅ id ) , using the Park transform. This equation is motivated further using Fig. 2.1.1 and Fig. 2.1.2. Torque is generated by the interaction between flux linkage and current, which is given by:. T = λ × i = λ ⋅ i sin(∠i − ∠λ ). (2.1.4). Equation (2.1.4) shows that if the current vector and flux linkage vector are displaced 90º spatially, then maximum torque can be expected. The direction of the torque is given by the thumb of the right hand if the other fingers of the right hand are curled from the flux linkage vector to the current vector. In Fig. 2.1.2, the D-axis flux linkage with the Q-axis current gives a positive torque (out of the page, towards the reader), while the Q-axis flux linkage with the D-axis current gives a negative torque (into the page, away from the reader). The sum of these torque components gives the total torque, i.e.. Ttotal = λ d ⋅ iq − λ q ⋅ id .. This derivation however, is for one D-axis and one Q-axis and therefore a one pole-pair machine. For a machine with p pole-pairs, the torque needs to be scaled with p. Furthermore, the power in the ABC reference frame must be equal to the power in the QD0 reference frame, which leads to the scaling. 3 2. 3 2. factor of ( ) in the QD0 reference frame. The torque equation should therefore by scaled by ( ) . The correct torque equation is thus given by:. 3 Tem = ( ) p(λ d ⋅ iq − λ q ⋅ id ) 2. (2.1.5). Equivalently, using magnitudes and angles in Fig. 2.1.2, the total torque is given by. Ttotal = λ s × i s = λ s ⋅ i s sin(φ − δ ). and it should be scaled to give the correct torque:. 13.

(25) Tem =. 3 p[λs ⋅ is sin(φ − δ )] . 2. Tem =. 3 p ⋅ λ s ⋅ i s (sin φ cos δ − cos φ sin δ ) . Taking the stator flux and stator current into the bracket: 2. Tem =. 3 p(i s sin φ ⋅ λ s cos δ − i s cos φ ⋅ λ s sin δ ) , which is equivalent to equation (2.1.5). 2. Expanding. this. equation. using. a. trigonometry. identity:. The speed voltage terms − ωe λq and + ωe λd in equations (2.1.2) and (2.1.3) can be explained physically using a torque concept. Three-dimensional views of the coils for the Q-axis and D-axis currents are shown in Fig. 2.1.3 and Fig. 2.1.4. q-axis. iq. T. iq. λd. λq d-axis. iq. −. iq. +. Vq. λd. T. Fig. 2.1.3. Q-axis coil in the QD0 representation.. q-axis. λq. T. id. id. λd. −. d-axis. id. λq. Vd. +. id. T. Fig. 2.1.4. D-axis coil in the QD0 representation.. 14.

(26) First consider the Q-axis coil, shown in Fig. 2.1.3. The applied Q-axis voltage causes a Q-axis current to flow, which is limited by the resistance of the coil, rs. This current in the coil causes a magnetic field that moves perpendicularly through the area spanned by the coil, in a direction given by the right hand rule. At the same time, the applied D-axis voltage causes a D-axis current to flow, and the current in turn causes a magnetic field that moves perpendicularly through the area spanned by the coil (Fig. 2.1.4).. In both Fig. 2.1.3 and Fig. 2.1.4, there is one D-axis and one Q-axis and therefore this represents a singe pole-pair machine for which the electrical speed and mechanical speed are the same. It is thus valid to say that the relationship between power and torque is given by P = T ⋅ ω e . Power can be expressed in terms of voltage and current, while torque can be expressed in terms of flux linkage and current. Therefore v ⋅ i = λ ⋅ i ⋅ ω e , and so v = λ ⋅ ω e .. The torque generated by the interaction of the D-axis flux linkage and the Q-axis current (Fig. 2.1.3) is in an anti-clockwise direction and is therefore positive according the vector diagram convention. The torque generated by the interaction of the Q-axis flux linkage and the D-axis current (Fig. 2.1.4) is in a clockwise direction and is therefore negative. This explains the signs of the speed voltage terms in equations (2.1.2) and (2.1.3). In a RSM with id = iq , note that λd >> λq due to the flux barriers. Therefore the positive torque shown in Fig. 2.1.3 is much larger than the negative torque shown in Fig. 2.1.4. By reducing the Qaxis flux linkage even further, the net positive torque will increase. This can be accomplished by introducing PMs that generate a constant flux linkage in the negative Q-axis direction. This explains why the PMs are placed inside the flux barriers, as shown in Fig. 2.1, and why they produce flux linkage in the negative Q-axis direction.. In equations (2.1.2) and (2.1.3), both D-axis and Q-axis flux linkages are functions of the D-axis current, Q-axis current and the electrical rotor position. Taking the Q-axis for example, the total differential dλ q is defined in terms of the differentials di d , diq and dθ e as follows [B16, p.947]:. dλ q =. ∂λ q. ∂id. did +. ∂λ q. ∂iq. diq +. ∂λ q. ∂θ e. dθ e. (2.1.6). Using the dash notation for partial derivatives, the partial derivatives of the flux linkage with respect to the currents can be written as self inductance and mutual inductance:. dλ q = M q' ⋅ did + L'q diq +. ∂λ q. ∂θ e. dθ e. (2.1.7). 15.

(27) From equation (2.1.7), the time derivative of the Q-axis flux linkage is given by:. dλ q dt. = M q'. diq ∂λ q dθ e did + L'q + dt dt ∂θ e dt. (2.1.8). If the rotor is skewed, the change in flux linkage due to rotor position is relatively small and can be omitted. The rotor of the PMA RSM however, is not skewed, because a skewed rotor means that the PMs have to be skewed as well. It is assumed in further analysis, that the change in flux linkage due to the change in rotor position is negligible. Equation (2.1.8) is thus approximated by:. dλ q dt. = M q'. diq did + L'q dt dt. (2.1.9). By analogy, the derivative of the D-axis flux linkage can be expressed as:. diq dλ d di = L'd d + M d' dt dt dt. (2.1.10). Substituting equations (2.1.9) and (2.1.10) into equations (2.1.2) and (2.1.3), the electrical model of the machine is complete and is given by equations (2.1.11) and (2.1.12).. vd = rs id + L'd. diq did + M d' − ω e λq dt dt. vq = rs iq + L'q. diq dt. + M q'. did + ω e λd dt. (2.1.11). (2.1.12). For a complete model it is necessary to include a mechanical model and then to combine the electrical and mechanical models. The mechanical equations are given by:. Tem = J eq. ωm =. dω m + β eq ω m + TL dt. ωe p. (2.1.13). (2.1.14). Equation (2.1.13) states that the produced electro-mechanical torque is equal to the sum of three terms. The first term takes into account the equivalent inertia, the second term accounts for the friction, and the third term is the load torque that is to be driven by the machine. Equation (2.1.14) relates the mechanical speed of the rotor ωm , to the speed at which the magnetic field rotates ωe , with the number of pole-pairs, p.. To connect the mechanical and electrical models, the torque equation, which is obtained using the equivalent circuit of the RSM and the power equation (see Appendix A), is used:. 16.

(28) Tem =. 3p (λd iq − λq id ) 2. (2.1.15). Equations (2.1.11) through to (2.1.15) represent the complete mathematical model for the RSM.. 2.2. The mathematical model for the PMA RSM. The mathematical model for the RSM is given by equations (2.1.11) through to (2.1.15). These equations need to be modified if PMs are added to the rotor. Simply substituting λq − λ pm in the place of λq is all that is necessary. Note that the flux linkage due to the permanent magnets is constant and the derivative thereof is zero. The motivation behind the use of permanent magnets is the effect they have on the applied voltage, produced torque, power factor and inner power factor, as will be pointed out in the text that follows.. The RSM voltage equations (2.1.11) and (2.1.12) are modified to the PMA RSM voltage equations by substitution:. v d = rs id + v q = rs iq +. dλ d − ω e (λ q − λ pm ) dt. d (λ q − λ pm ) dt. + ωe λd. (2.2.1). (2.2.2). Since the flux linkage due to the PMs is constant, equation (2.2.2) reduces to:. v q = rs iq +. dλ q dt. + ω e λd. (2.2.3). From equations (2.2.1) and (2.2.3), the PMA RSM voltage equations are thus:. v d = rs id +. ∂λ d did ∂λ d diq + − ω e (λq − λ pm ) ∂id dt ∂iq dt. v q = rs iq +. ∂λ q did ∂λ q diq + + ω e λd , ∂id dt ∂iq dt. which can be written as:. v d = rs id + L'd v q = rs iq + L'q. diq did + M d' − ω e (λ q − λ pm ) dt dt diq dt. + M q'. did + ω e λd dt. (2.2.4). (2.2.5). 17.

(29) From equation (2.2.4), the PMs reduce the D-axis speed voltage. This results in a reduced D-axis supply voltage and thus a reduced stator supply voltage, since v s =. v d2 + v q2 . It is this reduction in. supply voltage that gives the PMA RSM better performance in the field weakening or high speed region, compared to the RSM. When the RSM reaches its rated speed, there is simply not enough voltage to force more current into the machine, even though more current might be available. For the PMA RSM with the same rated speed as the RSM, the reduced supply voltage means that there is now a surplus amount of voltage that can be used to force more current into the PMA RSM.. The RSM torque equation (2.1.15) is modified to the PMS RSM torque equation as follows:. Tem =. 3 p (λd iq − (λq − λ pm )⋅ id ) 2. 3 p (λd iq − λqid + λ pmid ) 2 3 3 2 sin( 2φ ) = p ( Ld − Lq ) ⋅ is + pλ pm ⋅ is cos(φ ) 2 2 2 =. (2.2.6). From the equation above it can be seen that the produced torque is increased by the added PMs. It is important to distinguish between the partial derivatives, which are called instantaneous inductances or transient inductances, and linearized inductances. For example the partial derivative L'd in equation (2.2.5) is an instantaneous self inductance, while Ld in equation (2.2.6) is a linearized self inductance. Fig. 2.2.1 shows an example of the D-axis flux linkage, instantaneous self inductance L'd and linearized self inductance Ld :. λd (id ). λd id. = Ld. ∂λd = L'd ∂id. λd. id Fig. 2.2.1. D-axis flux linkage, instantaneous inductance and linearized inductance.. 18.

(30) To summarize, the equations that describe the PMA RSM are given as follows:. diq did + M d' − ω e (λq − λ pm ) dt dt. vd = rs id + L'd vq = rs iq + L'q. Tem =. 2.3.. dt. did + ω e λd dt. + M q'. (2.2.2). 3 p (λd iq − (λq − λ pm )⋅ id ) 2. Tem = J eq. ωm =. diq. (2.2.1). (2.2.6). dω m + β eq ω m + TL dt. (2.1.13). ωe. (2.1.14). p. Power factor and inner power factor. The steady state vector diagram below shows the vectors for the RSM (solid lines) and PMA RSM (red dotted lines).. Q-axis. Vs I s rs. Vs '. Vq. I s rs. jωλs. Is. jωλs '. Iq. λq θ λq '. Vd. Vd '. θ'. γ γ'. λs. λs '. φ Id. λd. D-axis. λ pm Fig. 2.6. Comparative steady state vector diagram.. 19.

(31) The power factor (PF) is defined as the cosine of the angle between the supply current and voltage vectors. Recall that P =| v s || i s | cos(θ ) , where θ is the PF angle and is between 0 and 90 degrees. It is shown in the steady state vector diagram of Fig. 2.6, that the PMs reduce the PF angle, which means an increased PF for the PMA RSM. If the angle is zero (PF = 1), then the reactive power associated with the machine is zero and therefore all the power form the source can be used to do real work, i.e. more output power. Given the same input power to the RSM and PMA RSM described here, the PMA RSM will give a larger power output and in that sense is more efficient than the RSM.. The inner power factor (IPF) is defined as the sine of the angle between the current vector and the flux vector.. This is an indication of the magnitude of the produced torque:. recall that. T = λ × i =| λ || i | sin(γ ) ,where γ is the IPF angle. The IPF angle is between 0 and 90 degrees. It is shown in the steady state vector diagram Fig. 2.6, that the PMs increase the IPF angle, which means an increased IPF.. 2.4.. Parameter values. For the models of the RSM and PMA RSM, many parameters have to be identified. These include the D-axis and Q-axis flux linkages, self inductances and mutual inductances. To calculate the flux linkages and inductances, FEA will be used. The stator winding resistance, number of pole pairs, equivalent inertia and equivalent friction coefficient will be considered next.. The number of pole pairs is chosen in the design of the machine and is known to be (p = 3). The friction coefficient is ideally zero, but is taken to be ( B = 0.1. Nm ). The equivalent inertia can rad / sec. be calculated by noting that the shaft and the rotor have the same mass-coefficient [B5, p.76]:. 1 J = [ M (rrotor ) 2 ] 2 2 M = pv = pπrrotor l stack length = 7850 ⋅ π (0.15 2 )0.34 = 188.66 kg. (2.4.1). 1 ⇒ J = [(188.66) ⋅ (0.15) 2 ] = 2.122 kg ⋅ m 2 2 The inertia is not a very important parameter in the design of the current and speed controllers, therefore the mass reduction due to flux barriers can be ignored.. 20.

(32) The stator winding resistance (per phase) is measured practically. The resistance between two phases, e.g. phase A and phase C is measured and then divided by two to give the per phase stator winding resistance. It was found to be: rs = 15mΩ. (2.4.2). The stator winding resistance can be calculated using an analytic formula given by [B2, p.91]. This analytic formula is used in the FEA program [P2] to calculate the per phase stator winding resistance for a temperature of 30ºC and the result is given by: rs = 15mΩ. (2.4.3). The FEA program generates files that give information about the machine, e.g. number of stator slots, number of conductors per slot etc. The FEA program and examples of the generated files can be found on the CDROM [P2].. The flux linkages can be obtained accurately using the FEA program. This is a software based solution where the physical dimensions and other information about the PMA RSM are entered into the FEA program. The program creates a mesh to partition the object into finite elements. For each element, the vector potential is determined and from all the vector potentials other attributes, like the flux linkages or torque, can be calculated. From the flux linkages it is possible to calculate self inductances and mutual inductances.. The FEA program described in [B2] was used to optimally design the RSM [A4][B17]. Then PMs were included in the design [B5]. The results shown in Fig. 2.4.1 (the next 2 pages) are obtained by using the same FEA program as in [B5]. From Fig. 2.4.1 it is possible to compare the parameters of the RSM and PMA RSM. The Matlab code for generating the graphs of Fig. 2.4.1 is given on the CDROM [F1].. On these graphs, note the demagnetizing effect of the mutual coupling:. as the Q-axis current. increases, the D-axis flux linkage decreases for the same value of D-axis current. This means the mutual inductances must be negative. In fact, where the deviation in the flux linkage is the greatest, the mutual inductance is at its negative maximum. Comparing the graphs for the RSM and PMA RSM, it seems that the demagnetizing effect is reduced by adding the permanent magnets. The demagnetizing effect is however very small.. The most obvious difference between the graphs for the RSM and PMA RSM is the Q-axis flux linkage. Note that λq < 0 for iq = 0 A . Graphs for the self inductances are obtained by numerically differentiating the data for the flux linkages, and that is why a result at zero current is not shown. The. 21.

(33) graphs for the mutual inductances are obtained by taking the average change in flux linkage, so that. M d' is a function of id , and M q' is a function of iq . An alternative method is shown in Appendix E.. Iq = 0 A. 0.8 0.6 0.4 Iq = 280 A 0.2 0. 0. 100 200 D-axis current [A]. -3. D-axis self inductance [H]. 8. x 10. 300. 4 Iq = 240 A 2. 0. 100 200 D-axis current [A]. 300. -1. 0. 0.4 Iq = 280 A 0.2 0. 100 200 D-axis current [A]. 300. 0. 100 200 D-axis current [A]. x 10. 300. D-axis self inductance for PMA RSM Iq = 0 A. 6 4 Iq = 240 A 2 0. 0. 100 200 D-axis current [A]. -4. D-axis mutual inductance for RSM. 0. -2. 0.6. -3. D-axis mutual inductance [H]. 1. x 10. Iq = 0 A. 0.8. 8. 6. 0. 1. D-axis self inductance for RSM Iq = 0 A. -4. D-axis mutual inductance [H]. D-axis flux linkage [Wb turns]. 1. D-axis flux linkage for PMA RSM. D-axis self inductance [H]. D-axis flux linkage [Wb turns]. D-axis flux linkage for RSM. 1. x 10. 300. D-axis mutual inductance for PMA RSM. 0. -1. -2. 0. 100 200 D-axis current [A]. 300. Fig. 2.4.1a. FEA results for D-axis flux linkages, self inductances and mutual inductances.. 22.

(34) 0.1 0. Id = 280 A. -0.1 -0.2 100 200 Q-axis current [A]. -3. Q-axis self inductance [H]. 4. x 10. 300. 2 1 Id = 240 A 100 200 Q-axis current [A]. -4. -0.1 -0.2 Id = 280 A. 100 200 Q-axis current [A]. 300. 100 200 Q-axis current [A]. x 10. 300. Q-axis self inductance for PMA RSM. 3 Id = 0 A 2 1 0. Q-axis mutual inductance for RSM. -1. 0. Id = 0 A 0. -3. 300. 0. -2. 0.1. 0. Q-axis mutual inductance [H]. 1. x 10. 0.2. 4. Id = 0 A. 0. 0.3. Q-axis self inductance for RSM. 3. 0. Q-axis flux linkage [Wb turns]. Id = 0 A. 0.2. 0. Q-axis mutual inductance [H]. Q-axis flux linkage for PMA RSM. Q-axis self inductance [H]. Q-axis flux linkage [Wb turns]. Q-axis flux linkage for RSM 0.3. 1. Id = 240 A 0. 100 200 Q-axis current [A]. 300. -4 x 10 Q-axis mutual inductance for PMA RSM. 0. -1. -2. 0. 100 200 Q-axis current [A]. 300. Fig. 2.4.1b. FEA results for Q-axis flux linkages, self inductances and mutual inductances.. 23.

(35) 2.5.. Method of current control for RSM and PMA RSM. In this subsection the author examines two different methods of current control for the RSM and PMA RSM, namely constant field current control (CFCC) and constant current angle control (CCAC). It is shown that for the RSM, both CFCC and CCAC are suitable, but for the PMA RSM, only CCAC is suitable.. The torque equation for the RSM is given by:. Tem =. 3p (λd iq − λq id ) 2. (2.5.1). Equation (2.5.1) can also be written in terms of linearized inductances:. 3 T = ( ) p( Ld ⋅ id ⋅ iq − Lq ⋅ iq ⋅ id ) 2 3 = ( ) p( Ld − Lq ) ⋅ id ⋅ iq 2. (2.5.2). If the D-axis current is kept constant, then equation (2.5.2) can be expressed as:. T = KT ⋅ iq. (2.5.3). From equation (2.5.3), the torque of the RSM can be controlled by increasing or decreasing the Q-axis current. This method, by which the D-axis current is kept constant and the torque is controlled by the Q-axis current, is known as CFCC, because the D-axis current is the “field current”. The D-axis current is reduced in the flux weakening or high speed region. Note that the right hand side of the QD0-plane is used, as illustrated in Fig. 2.5.1. Q-axis. is. iq φ. D-axis. id. Fig. 2.5.1. Constant field current control: the area used in the QD0-plane.. 24.

(36) To explain CCAC, equation (2.5.2) can be expressed as:. Tem =. 3 2 sin(2φ ) p( Ld − Lq ) ⋅ is 2 2. (2.5.4). By keeping the current angle constant in equation (2.5.4), the torque is controlled by increasing and decreasing the current magnitude. Note that now both the D-axis and Q-axis currents change with changing current magnitude. For CCAC, the top half of the QD-plane is used. Positive torque is generated by the current vector in the first quadrant and negative torque is generated by the current vector in the second quadrant, as illustrated in Fig. 2.5.2. Q-axis. is −. is +. φ−. id −. iq. φ+ id +. D-axis. Fig. 2.5.2. Constant current angle control: two possible current vectors.. CFCC was compared to CCAC in terms of response time of the currents and also in terms of the energy efficiency in Chapter 1. It was stated that CFCC gives a faster current response compared to CCAC, but CFCC is less energy efficient compared to CCAC. It is shown next that both CFCC and CCAC are suitable for the RSM, but only CCAC is suitable for the PMA RSM.. Using FEA [F2], it is possible to calculate the torque as a function of current angle, by keeping the current magnitude constant. Here the stator current is kept at the rated value: i s = 283 A . In Fig. 2.5.3 below, the torque is shown as a function of current angle in two different ways. The graph on the left is a Cartesian plot, while the graph on the right is a polar plot. On the polar plot the torque magnitude is given by the radius, and the current angle is the angle taken from the positive horizontal axis in the anti-clockwise direction. The polar plot is on a complex plane.. 25.

(37) RSM torque versus current angle 800 Q-axis 600 90 120. 400. Torque [Nm]. 500. 150. 200. 1000 60 30 D-axis 0. 180. 0 -200. 210. -400. 330 240. 300 270. -600 -800. 0. 100 200 300 Current angle [degrees]. 400. Fig. 2.5.3. Torque versus current angle for RSM.. On the Cartesian plot, it can be seen that the torque is positive for current angles from 0º to 90º. This region corresponds to the first quadrant of the polar plot. The torque locus corresponding to current angles from 0º to 90º is drawn in the first quadrant, since the torque is positive. The torque is negative for current angles from 90º to 180º. This region corresponds to the second quadrant of the polar plot. In the complex plane, − T∠φ = T∠(φ + 180°) , where T > 0 . Therefore, the torque locus for the current angles from 90º to 180º, finds itself in the forth quadrant, not the second quadrant.. The torque is positive for current angles from 180º to 270º. This corresponds to the third quadrant on the polar plot, and since the torque is positive, the torque locus finds itself in the third quadrant. The torque is negative for current angles from 270º to 360º. This corresponds to the forth quadrant on the polar plot. The torque locus on the polar plot for current angles from 270º to 360º finds itself in the second quadrant, because the torque is negative.. From Fig. 2.5.3, note that the maximum positive torque is produced with a current angle of 54º or (180+54)º and maximum negative torque is produced for a current angle of (180-54)º or (360-54)º. Other current angles therefore produce less torque; that is why CFCC does not always adhere to the maximum torque per ampere criteria.. 26.

(38) Since the torque locus of the RSM is symmetrical in the right hand half of the complex plane, CFCC is a suitable method of current control. Also, since the torque locus of the RSM is symmetrical in the top half of the complex plane, CCAC is a suitable method of current control.. The torque as a function of current angle for the PMA RSM is shown in Fig. 2.5.4. Again, the current magnitude is kept constant: i s = 283 A . PMA RSM torque versus current angle 1000 Q-axis 800 600. 90 120. 400. 500. 150 Torque [Nm]. 1000 60 30. 200 D-axis 0. 180. 0 -200. 330. 210 -400 240 -600. 300 270. -800 -1000. 0. 100 200 300 Current angle [degrees]. 400. Fig. 2.5.4. Torque versus current angle for PMA RSM.. When mapping the Cartesian plot on the left to the polar plot on the right, it should be noted that a negative torque is represented on the polar plot by a positive torque with an additional current angle of 180º, as explained previously. It can be seen from Fig. 2.5.4 that there is a large positive torque in the first quadrant and a large negative torque in the second quadrant, followed by a small positive torque in the third quadrant and a small negative torque in the forth quadrant. After the polarity of the different parts of the torque locus have been considered, the absolute value of the torque versus the current angle can be shown as in Fig. 2.5.5.. 27.

(39) Q-axis 90. 1000. 120. 60 800 600 30. 150 400 200 180. 0. 210. D-axis. 330. 240. 300 270. Fig. 2.5.5. Absolute value of torque vs. current angle for PMA RSM.. Since the torque locus of the PMA RSM is not symmetrical in the right hand half of the complex plane, CFCC is not a suitable method for current control. If CFCC is used for the PMA RSM, the machine will function well as a motor, but very poorly as a generator. The torque locus of the PMA RSM is symmetrical in the top half of the complex plane and therefore CCAC is a suitable method of current control. If CCAC is used for the PMA RSM, the machine functions well as a motor and a generator.. The torque equation of the PMA RSM also shows that CFCC would not be a suitable method to control the machine. The torque equation of the PMA RSM is given by:. Tem =. 3 p (λ d i q − λ q i d + λ pm i d ) 2. (2.5.5). Equation (2.5.5) can also be written as:. Tem =. 3 3 p( Ld − Lq ) ⋅ id iq + pλ pmid 2 2. (2.5.6). If the D-axis current is kept constant, equation (2.5.6) can be expressed as:. Tem = KT iq + K pm. (2.5.7). From equation (2.5.7), there is always a constant positive torque generated by the constant positive Daxis current and constant PM flux linkage and therefore the negative torque is smaller for CFCC. By using CCAC, with a current angle of φ = 54° , the machine produces maximum torque per ampere and is therefore more efficient. CCAC uses the top half of the QD0-plane and is suitable for both RSM and PMA RSM. In other words the Q-axis current is always kept positive, while the D-axis. 28.

(40) current traverses the entire D-axis. Since the D-axis current has a slower response time compared to the Q-axis current, CCAC is slower than CFCC. The tradeoff is therefore energy efficiency vs. response time. It can also be seen from the graphs that the PMA RSM has a maximum torque of 812 Nm, while the RSM only reaches 700 Nm. This verifies the earlier theoretical development.. From this discussion it is evident that only CCAC is an option for current control of the PMA RSM. It is the objective to have fast dynamic performance from the PMA RSM and this can be achieved using a load torque observer with compensation current feedback (Chapter 5). First, a machine model is built in simulation followed by current controllers and a basic speed controller. Only then can a load torque observer be investigated.. 29.

(41) Chapter 3. Design methodology for the current and speed controllers. In this chapter, the design methodology for the current and speed controllers for the PMA RSM is presented. Background theory about digital control systems is given and the theory is applied to design the controllers. Throughout the chapter various Matlab commands and tools are used to support the theory.. 3.1.. Different methods to design a digital controller. Many methods exist to design a controller. A serious issue is that the controller is digital. The implication is that the output of the digital controller has to be transformed from a discrete signal to a continuous signal. This is usually accomplished by a sample and hold element at the output of the digital controller. The “hold” means that the sampled value of the controller output is held constant for one sampling period (assuming a zero order hold). The zero order hold (ZOH) introduces a delay of half the sampling period (see Appendix B).. Design by emulation requires that the sampling time is sufficiently small (the sampling frequency should be at least ten times the chosen closed loop bandwidth to avoid inaccuracies). The design for the controller takes place in the continuous-time domain with an S-plane representation, and it is then transformed to a discrete-time representation in the Z-plane using one of many transformations (examples are Backward Euler, Tustin’s method or Matched pole-zero [B9, p.658][B7, p.330]). Note that there is an approximation when this method is used, because it is assumed that the delay introduced by the ZOH is negligible.. Another method is to design the controller completely in the Z-plane [B9, p.668] by translating both the plant and the ZOH to their discrete equivalents (see Appendix B). There is no approximation when using this method, because the time delay is accounted for.. The discrete equivalent of the ZOH and plant can be transformed to the W-plane using a bilinear. T w 2 [B7, p.393] . This transformation will be called the W-transform. The transformation, z → T 1− w 2 1+. design of the controller can then take place in the W-plane [B7, p.400]. The W-plane is very much like the S-plane, except that the primary strip for frequency in the S-plane, which is. 30.

(42) −. jω s jω s < jω < , ( s = jω ) is mapped to − ∞ < jv < ∞ , ( w = jv ) in the W-plane. This means 2 2. that the frequency scale in the W-plane is distorted. The whole idea of using the W-plane instead of the S-plane is that the effect of the ZOH is included, and frequency response techniques (Bode plot) can be used. When the design of the controller is completed in the W-plane, it is transformed back to the Z-plane using the inverse W-transform w →. 2 z −1 . T z +1. A combination of the methods discussed in this subsection will be used to design the current and speed controllers for the PMA RSM. First the current controllers are discussed and for that the electrical model of the PMA RSM should be considered.. 3.2.. Decoupling procedure for the electrical model. The equations for the electrical model of the PMA RSM are coupled: there is a Q-axis term in the Daxis equation and vice versa. A decoupling procedure is suggested to simplify the design of the current controllers. The electrical model for the PMA RSM is given in subsection 2.2 and is shown here again for convenience:. vd = rs id + L'd vq = rs iq + L'q. diq did + M d' − ω re (λq − λ pm ) dt dt diq dt. + M q'. did + ω re λd dt. (3.2.1) (3.2.2). The block diagram that represents equations (3.2.1) and (3.2.2) is shown in Fig. 3.2.1. Note that this block diagram is drawn in Matlab’s Simulink. Inputs and output are indicated by numbered capsules. Unfortunately the Greek alphabet is not available in Simulink and therefore the electrical rotor speed is written as Wre, the D-axis mutual inductance as Md, the Q-axis flux linkage as Q_flux etc. Also note that the instantaneous inductance L'd is written as Ld in the simulation diagram and Ld ≠ Ld , where. Ld is the linearized inductance. Note that Fig. 3.2.1 is just a representation of the equations (3.2.1) and (3.2.2); the block diagram is not used for simulation.. 31.

(43) Fig. 3.2.1. Coupled electrical model of the PMA RSM.. From the block diagram in Fig. 3.2.1 it is evident that to control the D-axis and Q-axis currents would be difficult, since they are coupled. A decoupling procedure is suggested in Fig. 3.2.2. The speed voltages and mutual inductance terms are subtracted by the controller where they are added in the machine and vice versa.. Fig. 3.2.2. Decoupling procedure for the electrical model.. This procedure results in decoupled differential equations represented by the block diagram shown in Fig. 3.2.3.. Fig. 3.2.3. Decoupled electrical model.. 32.

(44) It is important to make the distinction between the real machine voltages, Vd and Vq , and the applied control voltages, Vd ' and Vq ' , in the current controllers. The relationship is given below:. Vd = Vd '+ M d. diq. Vq = Vq '+ M q. did + ω re λd dt. dt. − ω re (λq − λ pm ). (3.2.3) (3.2.4). It will become clear where this procedure fits in when the design of the current controllers are complete. The open loop response is investigated next to show why it is necessary to use closed loop control.. 3.3.. Open loop current response. Equipped with mathematically decoupled models for the D-axis plant and the Q-axis plant (shown in Fig. 3.2.3), it is now possible to evaluate the open loop current response of the machine. Generally the open loop response of any plant is very slow – this means that high frequency inputs to the plant are heavily attenuated and thus cannot serve as significant control signals. It is equivalent to state that the bandwidth of the plant is small. The bandwidth can easily be determined by looking at a Bode diagram or a root locus for the plant.. For example, the decoupled D-axis plant is given by:. id 1 = ' v d Ld ⋅ s + rs. (3.3.1). Suitable parameters for the D-axis plant (see subsection 2.4) are rs = 15 mΩ and L'd = 2 mH . If these values are substituted into equation (3.3.1), the open loop D-axis plant has a single pole at. − rs. L'd. = −7.5 . The open loop Bode plot and step response for this plant is shown in Fig. 3.3.1.. 33.

(45) Bode Diagram 40. Magnitude (dB). 30. System: G Frequency (rad/sec): 7.05 Magnitude (dB): 33.7. 20 10 0. Phase (deg). -10 0. -45. -90 -1. 0. 10. 1. 10. 2. 10. 3. 10. 10. Frequency (rad/sec) Step Response. 70. System: G Settling Time (sec): 0.522. 60. Amplitude. 50. 40. 30. 20. 10. 0. 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. Time (sec). Fig. 3.1.1. Bode plot (top) and step response (bottom) for open loop D-axis plant.. From Fig. 3.1.1 it can be seen that the bandwidth of the open loop D-axis plant is 7 rad/sec (recall that the pole position is at -7.5). A small bandwidth means a slow response to a step input: τ s = 0.522 sec in this case. A slow current response cannot be tolerated in the control of the PMA RSM. It is therefore necessary to use closed loop current control. The general idea is to design a controller to generate a control input for the plant. The controller and plant are in a closed loop with feedback from the plant to the controller.. 34.

(46) 3.4.. Closed loop current response. Consider the block diagram in Fig. 3.4.1 which describes a closed-loop control system for the D-axis part of the decoupled electrical machine equations.. Fig. 3.4.1. D-axis current control system. For example, let the controller be a constant gain: D ( s ) = 10 . The closed loop transfer function is then H cl ( s ) =. 10 . With reference to the chosen parameters for the open loop plant in Ld s + rs + 10. subsection 3.3, the closed loop Bode plot and step response is shown in Fig. 3.4.2. Bode Diagram 0. Magnitude (dB). -5. System: Hcl Frequency (rad/sec): 5.03e+003 Magnitude (dB): -3.08. -10 -15 -20 -25. Phase (deg). -30 0. -45. -90 0. 1. 10. 2. 10. 3. 10. 4. 10. 5. 10. 10. Frequency (rad/sec) Step Response 1 System: Hcl Settling Time (sec): 0.000781. 0.9 0.8 0.7. Amplitude. 0.6 0.5 0.4 0.3 0.2 0.1 0. 0. 0.2. 0.4. 0.6 Time (sec). 0.8. 1. 1.2 -3. x 10. Fig. 3.4.2. Bode plot (top) and step response (bottom) for closed loop D-axis plant and controller.. 35.

(47) From the closed loop Bode plot it can be seen that the bandwidth of the closed loop system is 5000 rad/sec. Furthermore, the settling time for a step input to the closed loop is τ s = 0.8 ms . Therefore the controller and plant in a closed loop can give the required performance needed for current control of the PMA RSM.. The above analysis was performed in the continuous-time domain. The next subsection addresses issues relating to the discrete nature of the controller and to stability.. 3.5.. The discrete nature of the controller. In reality, the controller will not be analogue and will not produce a continuous output signal. The reference current will also be a discrete signal, which will be generated within a digital signal processor (DSP). There are many advantages when using digital controllers instead of analogue controllers. A control system that contains both discrete and continuous parts is called a sampled data system [B15, p.1]. The sampled data system for the D-axis current controller (discrete) and D-axis plant (continuous) can be represented by the block diagram in Fig. 3.5.1. 1 Idref (dig). D(z). DAC. 1. Vd'. Ld.s+rs. 1 Id (actual). 2 clock Id (dig). ADC. Fig. 3.5.1. D-axis sampled data current control system.. In Fig. 3.5.1 there are clearly an analogue section and a digital section. The digital controller is given by D(z) and the continuous plant model by G ( s ) =. 1 . The DSP produces a reference current. L s + rs ' d. This “current” is just a number representing the real reference current. The number representing the reference current is compared to a number representing the measured machine current. The difference between these numbers is the control input for the digital current controller. The output of the digital current controller is a number that represents the control voltage for the machine.. The function of the analogue-to-digital converter (ADC) and the digital-to-analogue converter (DAC) is to translate between the numbers representing the currents or voltages and the actual currents and voltages.. 36.

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